Gurland's ratio for the gamma function

Authors:
M. Merkle
Affiliations:
-
Venue:
Computers & Mathematics with Applications
Year:
2005

Citing 3
Cited 2

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Improved asymptotic formulas for the gamma function

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Abstract

We consider the ratio T(x, y) = @?(x)@?(y) / @?^2((x + y)/2) and its properties related to convexity, logarithmic convexity, Schur-convexity, and complete monotonicity. Several new bounds and asymptotic expansions for T are derived. Sharp bounds for the function x - x/(1 - e^-^x) are presented, as well as bounds for the trigamma function. The results are applied to a problem related to the volume of the unit ball in R^n and also to the problem of finding the inverse of the function x - T(1/x, 3/x), which is of importance in applied statistics.